Indeed, plenty of Catholics use contraception, have sex before marriage, and so on ... the old rules for living are largely not followed or taken in any way seriously by most Catholics these days.
I wouldn't say religiously-affiliated groups don't claim such powers. My cousin works for a Catholic school and fell pregnant a few months before her wedding. She was threatened with losing her job, until the powers-that-be (the priest) made the "kind" decision to let her stay since she would be married before the baby was born. But if that weren't the case? No job to return to after maternity leave! Another story from one of her female co-workers: her husband was beating her, so she planned to divorce him. But no, Catholicism cannot have that! She was told if she left her husband they would fire her, because Catholics do not believe in divorce. That's right, they actually prefer a woman to be beaten by her husband than to divorce him and get out of that dangerous environment. She stayed on and did not get a divorce for fear of losing her job.
This kind of stuff happens all the time in first-world, allegedly secular countries, and it is bullshit of the highest order.
Why is Obama's contraception thingo such a big drama? Let it be offered to everyone as a state initiative in all institutions. If they don't agree with contraception personally, guess what ... THEY STILL HAVE THE POWER TO SAY NO. Problem solved.